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Hey I am supposed to construct two sequences $(a_n)_{1}^{\infty}$ and $(b_n)_{1}^{\infty} $ , so that $ \lim_{n\rightarrow\infty } \frac{a_{n}}{b_{n}} \neq \lim_{n\rightarrow\infty } \frac{a}{b_{n}} $, where $\lim_{n\rightarrow\infty } a_n = a$. I tried many different combinations with 1/n, but nothing worked so far. Thanks!

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How about $a_n=\frac{1}{n}$ and $b_n=\frac{1}{n^2}$? Actually, $a_n=b_n=\frac{1}{n}$ works as well...

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Take $a_n = e^{-n}$, so that $a=0$, and $b_n = e^{-n}$. The first limit is then $1$, whereas the second is $0$.

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  • $\begingroup$ This a nicer example than mine, since both limits are finite. $\endgroup$ – Mars Plastic Feb 9 at 18:36
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    $\begingroup$ I do not see how it is nicer than yours, but thanks anyway. $\endgroup$ – Gibbs Feb 9 at 18:36
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You can just take $a_n=b_n=1/n$. The left hand side is 1 and the right hand side is zero.

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